P-moment exponential stability of Caputo fractional differential equations with impulses at random times and fractional order q ∈ (1, 2)

The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied.

Download Full-text

Non-instantaneous impulses in Caputo fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2017-0032 ◽

2017 ◽

Vol 20 (3) ◽

Cited By ~ 4

Author(s):

Ravi Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Real World ◽

Existence Results ◽

Advantages And Disadvantages ◽

Fractional Order Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

AbstractRecent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

Download Full-text

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽

10.2298/fil1716217a ◽

2017 ◽

Vol 31 (16) ◽

pp. 5217-5239 ◽

Cited By ~ 5

Author(s):

Ravi Agarwal ◽

Snehana Hristova ◽

Donal O’Regan

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Lyapunov Functions ◽

Sufficient Conditions ◽

Time Intervals ◽

Dini Derivative ◽

Comparison Results ◽

Strict Stability ◽

Caputo Fractional Differential Equations ◽

Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

Download Full-text

Implicit nonlinear fractional differential equations of variable order

Boundary Value Problems ◽

10.1186/s13661-021-01540-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Amar Benkerrouche ◽

Mohammed Said Souid ◽

Kanokwan Sitthithakerngkiet ◽

Ali Hakem

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Fractional Differential Equations ◽

Boundary Value ◽

Variable Order ◽

Stability Of Solutions ◽

Nonlinear Fractional Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

Download Full-text

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0030 ◽

2020 ◽

Vol 21 (6) ◽

pp. 571-587 ◽

Cited By ~ 2

Author(s):

Akbar Zada ◽

Sartaj Ali ◽

Tongxing Li

Keyword(s):

Differential Equations ◽

Fractional Order ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Integer Order ◽

New Class ◽

Proposed Model ◽

Fractional Differential ◽

Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

Download Full-text